Paramagnetism

Magnetic fieald in a medium. The field gets stronger,
due to the present magnetic moment. Magnetic moments would be fully
aligned, but thermic effects. Boltzmann factor. This is paramagnetic response.
Diamagnetism: the medium increased energy, the field must decrease.
Relation between angular and magnetic moment. Spin and
normal magnetic moment different proportionality - Lande factor.

Evaluation of the average Jz
will give the average contribution to the magnetization.
The sums are like sum of power series. The average
value can be obtained by differentiating the sum
of the power series. For J=1/2 very simple result.
x is the variable containing the exponent of
the Boltzmann expression.

The magnetization characterized by chi.
To obtain a reasonable expression, The Brillouin function
BJ(x) must be Taylor-expanded

Evaluation of chi in detail, lower part. Above:
from before

The estimate of chi.
Starting Ferromagnetism.

.....Starting Ferromagnetism. This is the origin
of the exchange interaction - details next time.